Solving For An Unknown Endpoint Using The MidpointThe Midpoint Of \[$\overline{RT}\$\] Is \[$S(0.5, -6.25)\$\]. One Endpoint Is \[$T(-5.5, 2.75)\$\]. What Are The Coordinates Of The Other Endpoint, \[$R\$\]?A.

Introduction

In geometry, the midpoint of a line segment is the point that divides the segment into two equal parts. Given the midpoint and one endpoint of a line segment, we can use the midpoint formula to find the coordinates of the other endpoint. In this article, we will use the midpoint formula to solve for an unknown endpoint of a line segment.

The Midpoint Formula

The midpoint formula is given by:

$$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two endpoints of the line segment.

Given Information

We are given the midpoint of the line segment $\overline{RT}$ as $S(0.5, -6.25)$ and one endpoint as $T(-5.5, 2.75)$. We need to find the coordinates of the other endpoint, $R$.

Using the Midpoint Formula

Let's use the midpoint formula to solve for the coordinates of $R$. We know that the midpoint of $\overline{RT}$ is $S(0.5, -6.25)$, so we can set up the equation:

$$\left(\frac{x_R+x_T}{2},\frac{y_R+y_T}{2}\right)=(0.5,-6.25)$$

where $(x_R, y_R)$ are the coordinates of $R$ and $(x_T, y_T)$ are the coordinates of $T$.

Solving for $x_R$ and $y_R$

We can solve for $x_R$ and $y_R$ by equating the corresponding coordinates:

$$\frac{x_R+x_T}{2}=0.5 \Rightarrow x_R+x_T=1$$

$$\frac{y_R+y_T}{2}=-6.25 \Rightarrow y_R+y_T=-12.5$$

We are given that $x_T=-5.5$ and $y_T=2.75$, so we can substitute these values into the equations:

$$x_R-5.5=1 \Rightarrow x_R=6.5$$

$$y_R+2.75=-12.5 \Rightarrow y_R=-15.25$$

Conclusion

Therefore, the coordinates of the other endpoint, $R$, are $(6.5, -15.25)$.

Example Use Case

This problem can be used to demonstrate the concept of the midpoint formula in a real-world scenario. For example, if we are given the coordinates of two points on a map, we can use the midpoint formula to find the coordinates of a third point that is equidistant from the two given points.

Step-by-Step Solution

1. Write down the midpoint formula: $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$
2. Substitute the given values into the formula: $\left(\frac{x_R+x_T}{2},\frac{y_R+y_T}{2}\right)=(0.5,-6.25)$
3. Equate the corresponding coordinates: $\frac{x_R+x_T}{2}=0.5$ and $\frac{y_R+y_T}{2}=-6.25$
4. Solve for $x_R$ and $y_R$ by substituting the given values of $x_T$ and $y_T$: $x_R-5.5=1$ and $y_R+2.75=-12.5$
5. Simplify the equations to find the values of $x_R$ and $y_R$: $x_R=6.5$ and $y_R=-15.25$

Tips and Variations

• To find the midpoint of a line segment, use the midpoint formula: $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$
• To find the coordinates of the other endpoint of a line segment, use the midpoint formula and the coordinates of the given endpoint and midpoint
• To find the distance between two points, use the distance formula: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
  Solving for an Unknown Endpoint Using the Midpoint: Q&A =====================================================

Introduction

In our previous article, we discussed how to use the midpoint formula to solve for an unknown endpoint of a line segment. In this article, we will answer some frequently asked questions about the midpoint formula and solving for an unknown endpoint.

Q: What is the midpoint formula?

A: The midpoint formula is a mathematical formula that is used to find the coordinates of the midpoint of a line segment. It is given by:

$$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two endpoints of the line segment.

Q: How do I use the midpoint formula to solve for an unknown endpoint?

A: To use the midpoint formula to solve for an unknown endpoint, you need to know the coordinates of the midpoint and one endpoint of the line segment. You can then substitute these values into the midpoint formula and solve for the coordinates of the other endpoint.

Q: What if I don't know the coordinates of the midpoint?

A: If you don't know the coordinates of the midpoint, you can use the midpoint formula to find the coordinates of the midpoint first. Then, you can use the midpoint formula again to solve for the coordinates of the other endpoint.

Q: Can I use the midpoint formula to find the distance between two points?

A: No, the midpoint formula is used to find the coordinates of the midpoint of a line segment, not the distance between two points. To find the distance between two points, you need to use the distance formula:

$$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$

Q: What if the line segment is vertical or horizontal?

A: If the line segment is vertical or horizontal, the midpoint formula is still applicable. However, you need to be careful when substituting the values into the formula, as the x-coordinates or y-coordinates may be the same.

Q: Can I use the midpoint formula to solve for an unknown endpoint in 3D space?

A: Yes, the midpoint formula can be extended to 3D space. However, you need to use the formula:

$$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right)$$

where $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ are the coordinates of the two endpoints of the line segment in 3D space.

Q: What are some real-world applications of the midpoint formula?

A: The midpoint formula has many real-world applications, such as:

• Finding the coordinates of a point that is equidistant from two given points
• Determining the midpoint of a line segment in a map or a graph
• Calculating the distance between two points in a 3D space
• Finding the coordinates of a point that lies on a line segment

Conclusion

In this article, we have answered some frequently asked questions about the midpoint formula and solving for an unknown endpoint. We hope that this article has provided you with a better understanding of the midpoint formula and its applications.

Example Use Case

Suppose we have two points, $A(2, 3)$ and $B(4, 5)$, and we want to find the coordinates of the midpoint of the line segment $\overline{AB}$. We can use the midpoint formula to find the coordinates of the midpoint:

$$\left(\frac{2+4}{2},\frac{3+5}{2}\right)=(3,4)$$

Therefore, the coordinates of the midpoint of the line segment $\overline{AB}$ are $(3, 4)$.

Step-by-Step Solution

1. Write down the midpoint formula: $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$
2. Substitute the given values into the formula: $\left(\frac{2+4}{2},\frac{3+5}{2}\right)$
3. Simplify the equation to find the coordinates of the midpoint: $(3, 4)$

Tips and Variations

• To find the midpoint of a line segment, use the midpoint formula: $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$
• To find the coordinates of the other endpoint of a line segment, use the midpoint formula and the coordinates of the given endpoint and midpoint
• To find the distance between two points, use the distance formula: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$